with a density matrix \(\rho _0 = \left|\psi _0\right>\left<\psi _0\right|\) . Since we're considering a system of two \(S=1/2\)  spins, the Hilbert space can be represented with a basis of four states. This means the density matrix and Hamiltonian can be written as 4x4 matrices. In the Fock-Liouville space, the density matrix becomes a 16 dimensional complex-valued vector, and the Liouvillian a 16x16 complex matrix that we can write as [1]}
\(\tilde{\mathcal {L}}_{\pm \pm } = i(\mathcal {H}_{\pm \pm }\otimes \mathbb {I}-\mathbb {I}\otimes \mathcal {H}_{\pm \pm }^T)\) 
